Higher Order Multiscale Finite Element Method for Heat Transfer Modeling
Abstract
:1. Introduction
- The specific properties, shapes [11] and weight/volume ratios of the constituents;
- The particle-based mesoscopic modeling based on a coarse-grained analysis, e.g., Monte Carlo method, lattice Boltzmann method [30];
1.1. Microscale Modeling
1.2. Mesoscale (Particle-Based) Modeling
1.3. Macroscale Modeling
2. Problem Formulation
- Dirichlet boundary conditions: on ;
- Neumann boundary conditions: on ( is the unit outward normal vector, denotes the heat flux across ).
3. Upscaling
3.1. Idea
Algorithm 1 Solve a heat transfer problem within a heterogeneous domain. |
Require: define the problem (heterogeneous domain and boundary conditions) Ensure: a coarse mesh and an appropriate refinement of each coarse element for n=1 to do {loop over coarse mesh elements} for m=1 to M do {loop over n-th element shape functions} solve local problem (5) in the n-th element for the m-th shape function end for compute and for the n-th element end for solve the coarse mesh problem using the effective matrices and vectors |
3.2. Formulation
- In 1D, we only solve the reduced Equation (5) (), obtaining the modified shape function. For the linear shape functions, we use 0 and 1 as the boundary conditions. For the “bubble” ones, is equal to zero. Exemplary standard and modified “bubble” shape functions for this case are shown in Figure 2. The horizontal thick lines represent the material distribution; thus, the standard shape function (Figure 2a) is the solution of problem (5) for the material with constant thermal conductivity in . The solution presented in Figure 2b was obtained using 50 finite elements, which comply with the microstructure schematically marked with the horizontal line. The green material is characterized by a thermal conductivity 10 times larger than the other one;
- In 3D, we need to solve reduced problems (5) along the edges and, subsequently, within the faces of the domain. Finally, Equation (5) is solved with the Dirichlet boundary conditions resulting from the lower scales auxiliary computations. A number of 3D-modified shape function examples for the linear elasticity problem can be found in [14,18].
3.3. Implementation
- For the periodic heterogeneous domains, we compute the effective stiffness matrix once, and use it for every coarse mesh element—the effective load vectors are different in most cases;
- For non-periodic heterogeneous domains, we can parallelize the computations of the coarse-mesh-element matrices and vectors and .
4. Numerical Results
4.1. Asphalt Concrete
4.2. Metal Foam
5. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Klimczak, M.; Cecot, W. Higher Order Multiscale Finite Element Method for Heat Transfer Modeling. Materials 2021, 14, 3827. https://doi.org/10.3390/ma14143827
Klimczak M, Cecot W. Higher Order Multiscale Finite Element Method for Heat Transfer Modeling. Materials. 2021; 14(14):3827. https://doi.org/10.3390/ma14143827
Chicago/Turabian StyleKlimczak, Marek, and Witold Cecot. 2021. "Higher Order Multiscale Finite Element Method for Heat Transfer Modeling" Materials 14, no. 14: 3827. https://doi.org/10.3390/ma14143827